Question

If \(f(x) = \dfrac{2x + 3}{3x - 2}\), \( x \neq \dfrac{2}{3} \), then \( f \circ f \) is{

• A. an even function
• B. not defined for all \( x \in \mathbb{R} \)
• C. a constant function
• D. an odd function

Hint:

If \( f(f(x)) = x \), then the function often exhibits symmetry. Always test \( f(-x) \) to check odd or even nature.

Answer 1

The Correct Option is D

Step 1: Find \( f(f(x)) \).
\[ f(f(x)) = f\left( \frac{2x+3}{3x-2} \right) \] Step 2: Substitute into the function.
\[ f(f(x)) = \frac{2\left( \frac{2x+3}{3x-2} \right) + 3}{3\left( \frac{2x+3}{3x-2} \right) - 2} \] Step 3: Simplify.
\[ f(f(x)) = \frac{7x}{7} = x \] Step 4: Test for odd function.
\[ f(f(-x)) = -x = -f(f(x)) \] Step 5: Conclusion.
Hence, \( f \circ f \) is an odd function.